There are bi-ideals of semigroups which are not quasi-ideals. In spite of this fact, here we show that a semigroup S is quasi-simple if and only if it is bi-simple, equivalently t-simple. Main results of this article are several equivalent characterizations of the Clifford semigroups and the semigroups which are semilattices of t-Archimedean semigroups by their quasi-ideals. A semigroup S is a Cli_ord semigroup if and only if every quasi-ideal of S is a semiprime ideal, whereas S is a semilattice of t-Archimedean semigroups if and only if p Q is an ideal for every quasi-ideal Q of S.